Which type of speech would you give if you wanted to convince your audience not to drop out of school?

A persuasive speech.

A persuasive speech is defined as a specific type of speech in which the speaker has a goal of convincing the audience to accept his or her point of view. Since what you're looking for is to convince your audience not to drop out of school, then this speech would be the most appropiate.

Definitely not an extemporaneous speech as this is carried out or performed with little or no preparation and that would not push your audience to want to stay in school. Not an entertaining speech because entertaining an audience is not the same as convincing them.

Therefore, you should go with a persuasive speech.